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Data StructuresLecture 4
Fang YuDepartment of Management Information SystemsNational Chengchi University
Fall 2010
Array ListThe Array List ADT extends the notion of array
by storing a sequence of arbitrary objects
An element can be accessed, inserted or removed by specifying its index (number of elements preceding it)
An exception is thrown if an incorrect index is given (e.g., a negative index)
Array ListMain methods:get(integer i): returns the element at index i
without removing it set(integer i, object o): replace the element at
index i with o and return the old element add(integer i, object o): insert a new element o to
have index i remove(integer i): removes and returns the
element at index i
Additional methods: size() isEmpty()
Array-based ImplementationUse an array A of size N
A variable n keeps track of the size of the array list (number of elements stored)
Operation get(i) is implemented in O(1) time by returning A[i]
Operation set(i,o) is implemented in O(1) time by performing t = A[i], A[i] = o, and returning t.
Insertion In operation add(i, o), we need to make room for
the new element by shifting forward the n - i elements A[i], …, A[n - 1]
In the worst case (i = 0), this takes O(n) time
A
0 1 2 ni
A
0 1 2 ni
A
0 1 2 noi
Element Removal In operation remove(i), we need to fill the hole
left by the removed element by shifting backward the n - i - 1 elements A[i + 1], …, A[n - 1]
In the worst case (i = 0), this takes O(n) time
A
0 1 2 ni
A
0 1 2 noi
A
0 1 2 ni
Performance In the array based implementation of an array
list: The space used by the data structure is O(n)
size, isEmpty, get and set run in O(1) time add and remove run in O(n) time in worst case
In an add operation, when the array is full, instead of throwing an exception, we can replace the array with a larger one
Growable Array-based Array ListIn an add(o) operation (without an index),
we always add at the end
When the array is full, we replace the array with a larger one
How large should the new array be?Incremental strategy: increase the size by
a constant cDoubling strategy: double the size
Growable Array-based Array List
Algorithm add(o)if t = S.length 1 then
A new array of
size …for i 0 to n-
1 do
A[i] S[i]S A
n n + 1S[n-1] o
//create a new array A (larger than S)
//copy the elements in S to A
//increase the size by 1//add o as the last element
//Replace S with A
Comparison of the StrategiesWe compare the incremental strategy and the
doubling strategy by analyzing the total time T(n) needed to perform a series of n add(o) operations
We assume that we start with an empty stack represented by an array of size 1
We call amortized time of an add operation the average time taken by an add over the series of operations, i.e., T(n)/n
Incremental Strategy We replace the array k = n/c times
The total time T(n) of a series of n add operations is proportional to
n + c + 2c + 3c + 4c + … + kc
Since c is a constant, T(n) is O(n + k2), i.e., O(n2)
The amortized time of an add operation is O(n)
Doubling Strategy We replace the array k = log2 n times
The total time T(n) of a series of n add operations is proportional to
n + 1 + 2 + 4 + 8 + …+ 2k =
3n - 1
T(n) is O(n)
The amortized time of an add operation is O(1)
geometric series
1
2
14
8
Singly Linked ListA concrete data structure
consisting of a sequence of nodes
Each node storeselement link to the next node
next
elem node
A B C D
The Node Class for Singly Linked List public class Node { private Object element; private Node next; public Node() { this(null, null); } public Node(Object e, Node n) { element = e; next = n; } public Object getElement() { return element; } public Node getNext() { return next; } public void setElement(Object
newElem) { element = newElem; } public void setNext(Node
newNext) { next = newNext; }}
// Instance variables
// Creates a node with null // references to its element and next node.
// Creates a node with the given element // and next node.
// Accessor methods
// Modifier methods
Inserting at the Head1. Allocate a new
node
2. Insert new element
3. Have new node point to old head
4. Update head to point to new node
Removing at the Head1. Update head to
point to next node in the list
2. Allow garbage collector to reclaim the former first node
Inserting at the Tail1. Allocate a new
node2. Insert new element3. Have new node
point to null4. Have old last node
point to new node5. Update tail to point
to new node
Removing at the TailRemoving at the tail
of a singly linked list is not efficient!
There is no constant-time way to update the tail to point to the previous node
Stack as a Linked List We can implement a stack with a singly linked list
The top element is stored at the first node of the list
The space used is O(n) and each operation of the Stack ADT takes O(1) time
t
nodes
elements
Queue as a Linked List We can implement a queue with a singly linked listThe front element is stored at the first nodeThe rear element is stored at the last node
The space used is O(n) and each operation of the Queue ADT takes O(1) time
21
f
r
nodes
elements
Position ADTThe Position ADT models the notion of place
within a data structure where a single object is stored
It gives a unified view of diverse ways of storing data, such as a cell of an array
a node of a linked list
Just one method: object element(): returns the element stored at
the position
Node List ADTThe Node List ADT models a sequence of positions storing
arbitrary objects
It establishes a before/after relation between positions
• Generic methods:• size(), isEmpty()
• Accessor methods:• first(), last()• prev(p), next(p)
• Update methods:• set(p, e) • addBefore(p, e), addAfter(p, e),• addFirst(e), addLast(e)• remove(p)
Doubly Linked ListA doubly linked list provides a natural
implementation of the Node List ADT
Nodes implement Position and store: element link to the previous node link to the next node
Special trailer and header nodes
prev next
elem node
InsertionWe visualize operation insertAfter(p, X), which returns
position q
A B X C
A B C
A B C
p
X
q
p q
Insertion Algorithm
Algorithm addAfter(p,e):Create a new node vv.setElement(e)v.setPrev(p) //link v to its predecessorv.setNext(p.getNext()) //link v to its successor(p.getNext()).setPrev(v)//link p’s old successor to vp.setNext(v) //link p to its new successor, vreturn v //the position for the element e
Deletion Algorithm
Algorithm remove(p):t = p.element //a temporary variable to hold the return value(p.getPrev()).setNext(p.getNext()) //linking out p(p.getNext()).setPrev(p.getPrev())p.setPrev(null) //invalidating the position pp.setNext(null)return t
HW4 (Due on 10/14)
Maintain an ordered keyword list.
A keyword is a pair [String name, Integer count]
Keep the list in order by its count while adding or deleting elements
For the list structure, you can Use java.util.ArrayList, or java.util.LinkedList, or Develop it by yourself
Given a sequence of operations in a txt file, parse the txt file and execute each operation accordingly
Add and Output
operations description
add(Keyword k) Insert k to the list in order
outputIndex(int i) Output the ith keyword in the list
outputCount(int c) Output all keywords whose count is equal to c
outputHas(string s) Output all keywords whose name contains s
outputName(string s) Output all keywords whose name is equal to s
outputFirstN(int n) Output the first n keywords
outputAll() Output the whole list
Output operations
Yu, 5 NCCU, 3 Fang, 3 UCSB, 2
…outputCount 3outputName YuoutputHas UoutputIndex 1outputFirstN 2outputAll…
[NCCU, 3] [Fang, 3][Yu, 5][NCCU, 3] [UCSB, 2][Yu, 5][Yu, 5] [NCCU,3][Yu, 5] [NCCU, 3] [Fang, 3] [UCSB, 2]
Delete
operations description
deleteIndex(int i) Delete the ith keyword in the list
deleteCount(int c) Delete all keywords whose count is equal to c
deleteHas(string s) Delete all keywords whose name contains s
deleteName(string s) Delete all keywords whose name is equal to s
deleteFirst(int n) Delete the first n keywords
Delete operations
Yu, 5 NCCU, 3 Fang, 3 UCSB, 2
deleteCount 3deleteName YudeleteHas UdeleteIndex 1deleteFirstN 2deleteAll
An input file
add Fang 3add Yu 5add NCCU 3add UCSB 2add MIS 2add Badminton 4add Food 3add Data 3add Structure 4outputAll deleteCount 3outputCount 2outputName YudeleteName YuoutputHas adeleteHas aoutputIndex 2deleteIndex 4deleteFirstN 1outputFirstN 3 deleteAll
1. You need to read the sequence of operations from a txt file2. The format is firm3. Raise an exception if the input does not match the format